Revisiting the evolution of bow-tie architecture in signaling networks

Bow-tie architecture is a layered network structure that has a narrow middle layer with multiple inputs and outputs. Such structures are widely seen in the molecular networks in cells, suggesting that a universal evolutionary mechanism underlies the emergence of bow-tie architecture. The previous theoretical studies have implemented evolutionary simulations of the feedforward network to satisfy a given input-output goal and proposed that the bow-tie architecture emerges when the ideal input-output relation is given as a rank-deficient matrix with mutations in network link intensities in a multiplicative manner. Here, we report that the bow-tie network inevitably appears when the link intensities representing molecular interactions are small at the initial condition of the evolutionary simulation, regardless of the rank of the goal matrix. Our dynamical system analysis clarifies the mechanisms underlying the emergence of the bow-tie structure. Further, we demonstrate that the increase in the input-output matrix reduces the width of the middle layer, resulting in the emergence of bow-tie architecture, even when evolution starts from large link intensities. Our data suggest that bow-tie architecture emerges as a side effect of evolution rather than as a result of evolutionary adaptation.


Supplementary Note 1. The ODE model for bow-tie architecture evolution.
To analyze evolutionary dynamics of the linear network model, we here consider a phenomenological ordinary differential equation (ODE) of the network that qualitatively mimics the evolutionary simulation described in the main text.The linear network model that is defined in the main text consists of L-1  ×  matrices { () … (%) } and describes a layered feedforward network with  nodes in each layer.The link intensities from node  in the  th layer to node  in the  + 1 th layer are described by  () as

)
The time evolution of the link intensity  '( (%) is given by eq.1 in the main text as a proxy of the evolutionary simulation and can be described in the follow matrix form: The evaluation function F is given as (2) Here, A is the total in-out relation matrix and given as The derivative of F by  '( (%) , -.
Supplementary Note 2. Bow-tie evolution when the goal matrix is full rank.
Here, we consider the 2 × 2 goal matrix, For considering the early stage of evolution starting from a small initial link intensity, we assume that  is much smaller than , and thus  −  ≈ −.The time evolution of link intensities is given as follows: The first column of the above equation can be rewritten as Since  $ ≤  ) ,  / ,  0 , the following inequality is obtained: Here we define  5 and R as follows.
By solving this, we have  (%) .The matrices  ($) and  ()) next to the network describe the interactions between each pair of layers.The transmitted value between each pair of layers can be described by a product of matrices as shown on the right side of the figure.

Appearance time of bow-tie architecture.
The lower bound of  5 () diverges within the finite time,  = Thus, the column that has a larger  5,6 diverges first.From this equation,  5 (t) is expected to diverge within a finite time.The divergence of  5,6 and  5,6 implies the emergence of bow-tie architecture (see Supplementary Fig.9).Namely, in the early phase of evolution, bow-tie architecture emerges.
Supplementary Fig.1.Schematics of the linear network model with 2 nodes × 3 layers.The activation level (i.e., expression level of the protein) is denoted near the node.The values are weighted by the link intensities and summed in the downstream node.In this network, the link intensities take a value of 0 or 1.The link intensity from node  in layer  to node  in layer  +1 is described by a  '(